One-Dimensional Site Response Analysis What do we mean? One-dimensional = Waves propagate in one direction only

One-Dimensional Site Response Analysis What do we mean? One-dimensional = waves propagate in one direction only Motion is identical on planes perpendicular to that motion to infinity to infinity

One-Dimensional Site Response Analysis What do we mean? One-dimensional = waves propagate in one direction only Motion is identical on planes perpendicular to that motion Can’t handle refraction so layer boundaries must be perpendicular to direction of wave propagation Usual assumption is vertically-propagating shear (SH) waves Horizontal surface motion Horizontal input motion

One-Dimensional Site Response Analysis When are one-dimensional analyses appropriate? Stiffer with depth Focus

Horizontal boundaries – waves tend to be refracted toward vertical One-Dimensional Site Response Analysis When are one-dimensional analyses appropriate? Horizontal boundaries – waves tend to be refracted toward vertical Stiffer with depth Decreasing stiffness causes refraction of waves to increasingly vertical path Focus

One-Dimensional Site Response Analysis When are one-dimensional analyses appropriate? Not appropriate here Stiffer with depth

Not here! One-Dimensional Site Response Analysis Retaining structures When are one-dimensional analyses appropriate? Not here! Retaining structures Dams and embankments Inclined ground surface and/or non-horizontal boundaries can require use of two-dimensional analyses Tunnels

Localized structures may require use of 3-D response analyses One-Dimensional Site Response Analysis When are one-dimensional analyses appropriate? Not here! Complex soil conditions Dams in narrow canyons Multiple structures Localized structures may require use of 3-D response analyses

Rock outcropping motion One-Dimensional Site Response Analysis How should ground motions be applied? Rock outcropping motion 2ui Free surface motion us Soil Bedrock motion ui + ur Not the same! Rock Incoming motion ui

One-Dimensional Site Response Analysis How should ground motions be applied? Free surface motion us Input (object) motion If recorded at rock outcrop, apply as outcrop motion (program will remove free surface effect). Bedrock should be modeled as an elastic half-space. If recorded in boring, apply as within-profile motion (recording does not include free surface effect). Bedrock should be modeled as rigid. Object motion

Methods of One-Dimensional Site Response Analysis Complex Response Method Approach used in computer programs like SHAKE Transfer function is used with input motion to compute surface motion (convolution) For layered profiles, transfer function is “built” layer-by-layer to go from input motion to surface motion Single elastic layer Amplification De-amplification

Amplitudes of upward- and downward-traveling waves in Layer j Complex Response Method (Linear analysis) Consider the soil deposit shown to the right. Within a given layer, say Layer j, the horizontal displacements will be given by Layer j Layer j+1 Amplitudes of upward- and downward-traveling waves in Layer j At the boundary between layer j and layer j+1, compatibility of displacements requires that No slip Continuity of shear stresses requires that Equilibrium satisfied

Complex Response Method (Linear analysis) Defining a*j as the complex impedance ratio at the boundary between layers j and j+1, the wave amplitudes for layer j+1 can be obtained from the amplitudes of layer j by solving the previous two equations simultaneously Propagation of wave energy from one layer to another is controlled by (complex) impedance ratio Wave amplitudes in Layer j Wave amplitudes in Layer j+1 So, if we can go from Layer j to Layer j+1, we can go from j+1 to j+2, etc. This means we can apply this relationship recursively and express the amplitudes in any layer as functions of the amplitudes in any other layer. We can therefore “build” a transfer function by repeated application of the above equations.

Complex Response Method (Linear analysis) Single layer on rigid base H = 100 ft Vs = 500 ft/sec x = 10%

Complex Response Method (Linear analysis) Single layer on rigid base H = 50 ft Vs = 1,500 ft/sec x = 10%

Complex Response Method (Linear analysis) Single layer on rigid base H = 100 ft Vs = 300 ft/sec x = 5%

Complex Response Method (Linear analysis)

Complex Response Method (Linear analysis)

Complex Response Method (Linear analysis) Different sequence of soil layers Different transfer function Different response

Complex Response Method (Linear analysis) Another sequence of soil layers Different transfer function Different response

Complex Response Method (Linear analysis) Complex response method operates in frequency domain Input motion represented as sum of series of sine waves Solution for each sine wave obtained Solutions added together to get total response Principle of superposition t g Linear system Can we capture important effects of nonlinearity with linear model?

x Equivalent Linear Approach Equivalent shear modulus Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. x Equivalent shear modulus Equivalent damping ratio

Assume some initial strain and use to estimate G and x Equivalent Linear Approach Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. x g(1) g(1) Assume some initial strain and use to estimate G and x

Use these values to compute response Equivalent Linear Approach Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. g(t) t x g(1) g(1) Use these values to compute response

Determine peak strain and effective strain Equivalent Linear Approach Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. g(t) t gmax geff x g(1) g(1) Determine peak strain and effective strain geff = Rg gmax

Select properties based on updated strain level Equivalent Linear Approach Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. x g(1) g(2) g(1) g(2) Select properties based on updated strain level

x g(1) g(3) g(2) g(1) g(3) g(2) Equivalent Linear Approach Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. x g(1) g(3) g(2) g(1) g(3) g(2) Compute response with new properties and determine resulting effective shear strain

x geff geff Equivalent Linear Approach Soils exhibit nonlinear, inelastic behavior under cyclic loading conditions Stiffness decreases and damping increases as cyclic strain amplitude increases The nonlinear, inelastic stress-strain behavior of cyclically loaded soils can be approximated by equivalent linear properties. x geff geff Repeat until computed effective strains are consistent with assumed effective strains

Equivalent Linear Approach Advantages: Can work in frequency domain Compute transfer function at relatively small number of frequencies (compared to doing calculations at all time steps) Increased speed not that significant for 1-D analyses Increased speed can be significant for 2-D, 3-D analyses Equivalent linear properties readily available for many soils – familiarity breeds comfort/confidence Can make first-order approximation to effects of nonlinearity and inelasticity within framework of a linear model The equivalent linear approach is an approximation. Nonlinear analyses are capable of representing the actual behavior of soils much more accurately. … often, a very good one!

Nonlinear Analysis Divide time into series of time steps t Equation of motion must be integrated in time domain Wave equation for visco-elastic medium Divide time into series of time steps t Divide profile into series of layers z

Nonlinear Analysis Divide time into series of time steps tj t zi Equation of motion must be integrated in time domain Wave equation for visco-elastic medium Divide time into series of time steps tj t zi Divide profile into series of layers vij = v (z = zi, t = tj) z

Nonlinear Analysis tj t Equation of motion must be integrated in time domain Wave equation for visco-elastic medium tj t More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. zi z

Nonlinear Analysis tj t Equation of motion must be integrated in time domain Wave equation for visco-elastic medium tj t More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. zi Can change material properties for use in next time step. Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response. z

Nonlinear Analysis tj t Equation of motion must be integrated in time domain Wave equation for visco-elastic medium tj t More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. zi Can change material properties for use in next time step. Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response. z

Nonlinear Analysis tj t Equation of motion must be integrated in time domain Wave equation for visco-elastic medium tj t More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. zi Can change material properties for use in next time step. Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response. z

Nonlinear Analysis tj t Equation of motion must be integrated in time domain Wave equation for visco-elastic medium tj t More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. zi Can change material properties for use in next time step. Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response. z

Nonlinear Analysis tj t Equation of motion must be integrated in time domain Wave equation for visco-elastic medium tj t More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile. zi Step through time Can change material properties for use in next time step. Changing stiffness based on strain level, strain history, etc. can allow prediction of nonlinear, inelastic response. Procedure steps through time from beginning of earthquake to end. z

t t g g Nonlinear Behavior Actual Approximation Continuous Linear segments In a nonlinear analysis, we approximate the continuous actual stress-strain behavior with an incrementally-linear model. The finer our computational interval, the better the approximation.

Nonlinear Approach Advantages: Work in time domain Can change properties after each time step to model nonlinearity Can formulate model in terms of effective stresses Can compute pore pressure generation Can compute pore pressure redistribution, dissipation Avoids spurious resonances (associated with linearity of EL approach) Can compute permanent strain permanent deformations Liquefaction Nonlinear analyses can produce results that are consistent with equivalent linear analyses when strains are small to moderate, and more accurate results when strains are large. They can also do important things that equivalent linear analyses can’t, such as compute pore pressures and permanent deformations.

Equivalent Linear vs. Nonlinear Approaches What are people using in practice? Equivalent linear analyses One-dimensional – 2-D / 3-D – SHAKE QUAD4, FLUSH Nonlinear analyses One-dimensional – 2-D / 3-D – DESRA, DMOD TARA, FLAC, PLAXIS

Equivalent Linear vs. Nonlinear Approaches What are people using in practice? Equivalent linear analyses One-dimensional – 2-D / 3-D – SHAKE QUAD4, FLUSH Nonlinear analyses One-dimensional – 2-D / 3-D – DESRA TARA

Since early 1970s, numerous computer programs developed for site Available Codes Since early 1970s, numerous computer programs developed for site response analysis Can be categorized according to computational procedure, number of dimensions, and operating system Dimensions OS Equivalent Linear Nonlinear 1-D DOS Dyneq, Shake91 AMPLE, DESRA, DMOD, FLIP, SUMDES, TESS Windows ShakeEdit, ProShake, Shake2000, EERA CyberQuake, DeepSoil, NERA, FLAC, DMOD2000 2-D / 3-D FLUSH, QUAD4/QUAD4M, TLUSH DYNAFLOW, TARA-3, FLIP, VERSAT, DYSAC2, LIQCA, OpenSees QUAKE/W, SASSI2000 FLAC, PLAXIS

Attendees at ICSDEE/ICEGE Berkeley conference (non-academic) Current Practice Informal survey developed to obtain input on site response modeling approaches actually used in practice Emailed to 204 people Attendees at ICSDEE/ICEGE Berkeley conference (non-academic) Geotechnical EERI members – 2003 Roster (non-academic) 55 responses Western North America (WNA) Eastern North America (ENA) Overseas Private firms Public agencies Survey Respondents WNA ENA Overseas Private Public Number of responses 35 3 6 1 5

Current Practice Method of Analysis Of the total number of site response analyses you perform, indicate the approximate percentages that fall within each of the following categories: [ ] a. One-dimensional equivalent linear [ ] b. One-dimensional nonlinear [ ] c. Two- or three-dimensional equivalent linear [ ] d. Two- or three-dimensional nonlinear Method of Analysis WNA ENA Overseas Private (35) Public (3) (6) (1) (5) 1-D Equivalent Linear 68 52 86 50 24 5 1-D Nonlinear 11 17 12 48 2-D/3-D Equiv. Linear 9 28 1 25 6 2-D/3-D Nonlinear 3 23 90 One-dimensional equivalent linear analyses dominate North American practice; nonlinear analyses are more frequently performed overseas

Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? u(0,t) 1 m 15 m 29 m Ts = 0.4 sec 30 m Vs = 300 m/sec u(H,t) Topanga record (Northridge) Vs = 762 m/sec Topanga record (Northridge)

Low degree of nonlinearity Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Topanga motion scaled to 0.05 g Weak motion + stiff soil Low strains Low degree of nonlinearity Similar response

Low degree of nonlinearity Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Topanga motion scaled to 0.05 g Weak motion + stiff soil Low strains Low degree of nonlinearity Similar response

Low degree of nonlinearity Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Topanga motion scaled to 0.05 g Weak motion + stiff soil Low strains Low degree of nonlinearity Similar response

Low degree of nonlinearity Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Topanga motion scaled to 0.05 g Weak motion + stiff soil Low strains Low degree of nonlinearity Similar response

Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Topanga motion scaled to 0.20 g Moderate motion + stiff soil Relatively low strains Relatively low degree of nonlinearity Similar response

Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Acceleration Topanga motion scaled to 0.20 g Moderate motion + stiff soil Velocity Relatively low strains Relatively low degree of nonlinearity Similar response

Nonlinear Behavior Equivalent linear overpredicts nonlinear response at certain frequencies – “spurious resonances” Equivalent linear vs nonlinear analysis – how much difference does it make? Topanga motion scaled to 0.20 g Moderate motion + stiff soil Relatively low strains Stress-strain response becoming more complicated – more variable stiffness and less “elliptical” shape Relatively low degree of nonlinearity Similar response

Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Topanga motion scaled to 0.20 g Moderate motion + stiff soil Relatively low strains Stiffness starting to vary more significantly over course of ground motion Relatively low degree of nonlinearity Similar response

Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Acceleration Topanga motion scaled to 0.50 g Strong motion + stiff soil Moderate strains Low – moderate degree of nonlinearity Noticeably different response

Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Topanga motion scaled to 0.50 g Strong motion + stiff soil Moderate strains Low – moderate degree of nonlinearity Noticeably different response

Softening by EL method causes underprediction Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Acceleration Topanga motion scaled to 1.0 g Substantial softening by EL method causes underprediction of initial portion of record Very strong motion + stiff soil Softening by EL method causes underprediction Linearity inherent in EL method causes overprediction response in strongest portion of record Moderate strains Moderate degree of nonlinearity Noticeably different response

Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Topanga motion scaled to 0.50 g Very strong motion + stiff soil Moderate strains Moderate degree of nonlinearity Noticeably different response

Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? u(0,t) 1 m 15 m 29 m 16 m Vs = 100 m/sec Vs = 300 m/sec 14 m u(H,t) Vs = 762 m/sec

Nonlinear Behavior Acceleration Equivalent linear vs nonlinear analysis – how much difference does it make? EL model predicts very soft behavior at beginning of earthquake, before any large strains have developed. Large strain levels (~6%) near bottom of upper layer EL model converges to low G and high x High-frequency components cannot be transmitted through over-softened EL model NL model: Stiffness stays relatively high except for a few large-amplitude cycles

Nonlinear Behavior Acceleration Equivalent linear vs nonlinear analysis – how much difference does it make? Large strain levels (~6%) near bottom of upper layer More consistency, but NL model can transmit high-frequency oscillations superimposed on low-frequency cycles – too much? EL model converges to low G and high x High-frequency components cannot be transmitted through over-softened EL model NL model: Stiffness stays relatively high except for a few large-amplitude cycles

Nonlinear Behavior Acceleration Equivalent linear vs nonlinear analysis – how much difference does it make? Large strain levels (~6%) near bottom of upper layer NL model exhibits stiff behavior following strongest part of record; EL maintains low stiffness, high damping behavior throughout. EL model converges to low G and high x High-frequency components cannot be transmitted through over-softened EL model NL model: Stiffness stays relatively high except for a few large-amplitude cycles

Nonlinear Behavior Equivalent linear vs nonlinear analysis – how much difference does it make? Large strain levels (~6%) near bottom of upper layer EL model converges to low G and high x High-frequency components cannot be transmitted through over-softened EL model NL model: Stiffness stays relatively high except for a few large-amplitude cycles

Nonlinear Soil Behavior Small cycle superimposed on large cycle (after Assimaki and Kausel, 2002) Time High stiffness Equivalent linear model maintains constant stiffness and damping – higher stiffness excursions associated with higher frequency oscillations aren’t seen. Low stiffness

Nonlinear Soil Behavior Small cycle superimposed on large cycle (after Assimaki and Kausel, 2002) Time Low damping Equivalent linear model maintains constant stiffness and damping – higher stiffness excursions associated with higher frequency oscillations aren’t seen. High damping

Modified Equivalent Linear Approach High frequencies are associated with smaller strains High stiffness and low damping are associated with smaller strains Make stiffness and damping frequency-dependent Normalized strain spectra from five motions Normalized strain spectrum from one motion Three orders of magnitude Frequency (Hz) Frequency (Hz)

Frequency-dependent model Modified Equivalent Linear Approach Assimaki and Kausel Frequency-dependent model Conventional model High frequencies oversoftened and overdamped Excellent agreement with nonlinear model

Benchmarking of Nonlinear Analyses Stewart and Kwok PEER study to determine proper manner in which to use nonlinear analyses Worked with five existing nonlinear codes; hired developers to run their codes and comment on results Established advisory committee to oversee analyses and assist with interpretation Met regularly with advisory committee and developers

Benchmarking of Nonlinear Analyses Stewart and Kwok Considered codes

Benchmarking of Nonlinear Analyses D-MOD_2 (Matasovic) Enhanced version of D-MOD, which is enhanced version of DESRA Lumped mass model Rayleigh damping Rayleigh Damping ratio Stiffness-proportional Mass-proportional Frequency

Benchmarking of Nonlinear Analyses D-MOD_2 (Matasovic) Enhanced version of D-MOD, which is enhanced version of DESRA Lumped mass model Rayleigh damping Newmark b method for time integration Variable slice width – simulating response of dams, embankments on rock Decreasing stiffness due to geometry

Benchmarking of Nonlinear Analyses D-MOD_2 (Matasovic) Enhanced version of D-MOD, which is enhanced version of DESRA Lumped mass model Rayleigh damping Newmark b method for time integration Variable slice width – simulating response of dams, embankments on rock Can simulate slip on weak interfaces Uses MKZ soil model (modified hyperbola – needs Gmax, tmax, a and s) Can soften backbone curve to model cyclic degradation

Benchmarking of Nonlinear Analyses D-MOD_2 (Matasovic) Enhanced version of D-MOD, which is enhanced version of DESRA Lumped mass model Rayleigh damping Newmark b method for time integration Variable slice width – simulating response of dams, embankments on rock Can simulate slip on weak interfaces Uses MKZ soil model (modified hyperbola – needs Gmax, tmax, a and s) Can soften backbone curve to model cyclic degradation Uses Masing rules for unloading-reloading behavior Need input parameters for: MKZ backbone curve (4) Cyclic degradation (3 for clay, 4 for sand) Pore pressure generation (4 for clay, 4 for sand) Pore pressure redistribution/dissipation (at least 2) Rayleigh damping coefficients (2) Basic layer properties (density, shear wave velocity, half-space properties)

Benchmarking of Nonlinear Analyses DEEPSOIL (Hashash) Similar to DMOD-2 (lumped mass, derives from DESRA-2) More advanced Rayleigh damping scheme (lower frequency dependence) TESS (Pyke) Finite difference wave propagation analysis (not lumped mass) Cundall-Pyke hypothesis for loading-unloading behavior Similar backbone curve to DMOD-2 and DEEPSOIL Inviscid (sort of) low-strain damping scheme OpenSees (Yang, Elgamal) Finite element model (1D, 2D, 3D capabilities) Multi-surface plasticity model (von Mises yield surface, kinematic hardening, non-associative flow rule) Full Rayleigh damping SUMDES Finite element model Bounding surface plasticity model (Lade-like yield surface, kinematic hardening, non-associative flow rule) Simplified Rayleigh damping

Benchmarking of Nonlinear Analyses Recommendations Specification of control motion For outcropping motion, use recorded motion with elastic base For motions recorded at depth, use recorded motion with rigid base Specification of viscous damping Use full or extended Rayleigh damping – iterate on selection of control frequencies to match equivalent linear response for low loading levels (linear response domain). If not possible, use full Rayleigh damping with targets at fo and 5fo. Backbone curve parameters Adjust, if possible, to produce correct shear strength at large strains Bound nonlinear, inelastic behavior by running analyses with: Backbone curve fit to match G/Gmax behavior Backbone curve fit to minimize error in G/Gmax and damping curves

Benchmarking of Nonlinear Analyses Performance Based on validations against vertical array data Models produce reasonable results Some indication of overdamping at high frequencies, overamplification at site frequency Variability of predictions due to backbone curves and damping models most pronounced at T<0.5 sec and is significant only for relatively thick profiles. Model-to-model variability most pronounced at low periods. Nonlinearity modeled well up to levels for which adequate data is available (generally up to about 0.2g). Data for stronger shaking being sought (centrifuge tests, recent Nigaata earthquake). DMOD-2, DEEPSOIL, and OpenSees generally produced similar amplification factors and spectral shapes; TESS produced different response at high frequencies (different damping formulation), SUMDES results were significantly different than all others for deep sites (probably due to simplified Rayleigh damping).

Nonlinear Behavior – Effective Stress Analyses Wildlife – Superstition Hills recordings

Nonlinear Behavior – Effective Stress Analyses Wildlife – Superstition Hills recordings

Nonlinear Behavior – Effective Stress Analyses Wildlife – Elmore Ranch recordings

Nonlinear Behavior – Effective Stress Analyses Wildlife – Superstition Hills recordings Ground surface record High frequency ??? Low frequency

Site Effects Elmore Ranch record – no liquefaction Ratio of wavelet amplitudes – variation with frequency and time Frequency (Hz) Time (sec)

Site Effects Elmore Ranch record – no liquefaction Ratio of wavelet amplitudes – variation with frequency and time Frequency (Hz) Time (sec)

Nonlinear Behavior – Effective Stress Analyses Wildlife – Superstition Hills recordings

Nonlinear Behavior – Effective Stress Analyses Wildlife – Superstition Hills recordings

One-Dimensional Site Response Analysis Summary Must be aware of assumptions Uni-directional wave propagation (normal to layer boundaries) Uni-directional particle motion (no surface waves) Particularly useful for profiles with high impedance contrasts Equivalent linear approach works very well for most cases Material properties readily available Computations performed rapidly Nonlinear analyses match equivalent linear when strains are small Nonlinear analyses are preferred when strains are high – soft soils and/or strong shaking Can account for shear strength of soil Can handle pore pressure generation – some well, some poorly Can predict permanent deformations – for common for 2-D analyses

Thank you